Question

Calculate the mean deviation from the mean for the following data:
$57,64,43,67,49,59,44,47,61,59$

Answer 1

Hint: Since they have asked to calculate the mean deviation using the mean of the given data , we will first find the mean of the given ungrouped data, then use the formula for mean deviation using mean value to find the required mean deviation.

Complete step-by-step answer:
The given ungrouped data values are $57,64,43,67,49,59,44,47,61,59$
The mean can be calculated by dividing the sum of all the observations by the total number of observations.
Let $\sum {{x_i}} $be the sum of all the observations where ${x_i}$ is the i’th observation and
$n = $Total number of observations then mean $M$ is given by,
$\overline x = \dfrac{{\sum {{x_i}} }}{n}$
$\overline x = \dfrac{{57 + 64 + 43 + 67 + 49 + 59 + 44 + 47 + 61 + 59}}{{10}}$
$ \Rightarrow \overline x = \dfrac{{550}}{{10}} = 55$.
Now the formula for mean deviation using mean is given by ,
$MD = \dfrac{{\sum\limits_{i = 1}^n {|{x_i} - \overline x |} }}{n}$, where$\sum\limits_{i = 1}^n {|{x_i} - \overline x |} $ is obtained by subtracting each observation from the calculated mean value and then taking the mod to avoid negative sign then taking its summation for which let us consider the following table,

Observations	$|{x_i} - \overline {x|} $
57	2
64	9
43	12
67	12
49	6
59	4
44	11
47	8
61	6
59	4
	$\sum\limits_{i = 1}^n {|{x_i} - \overline x |} = 74$

$ \Rightarrow MD = \dfrac{{\sum\limits_{i = 1}^n {|{x_i} - \overline x |} }}{n} = \dfrac{{74}}{{10}} = 7.4$
Therefore the required mean deviation is $7.4$
So, the correct answer is “7.4”.

Note: The mean deviation can be calculated using three different methods with the help of mean, median or mode. In the question if they don’t mention the specific method then you can use any of the three different methods of mean, median or mode which ever seems easier for you.